We ¯nd that when the hypothetical benchmarks are recognized as being unavailable for investment, there need not exist close substitutes for them in the universe of mutual funds. For an investor who believes completely in the accuracy of the Fama-French model and precludes managerial skill, the perceived maximum Sharpe ratio is only 66 percent of what could be achieved by direct investment in that model's benchmarks. For a believer in the Carhart four-factor model, the corresponding value is 54 percent. Moreover, actively managed funds can be better substitutes for the benchmarks than existing passive funds, so active funds can be selected even by investors who admit no.
